A discussion of the relativistic equal-time equation

Ruan Tu-nan et al. [1] have proposed an equal-time equation for composite particles which is derived from Bethe-Salpeter (B-S) equation. Its advantage is that the kernel of this equation is a completely definite single rearrangement of the B-S irreducible kernel without any artificial assumptions. In this paper we shall give a further discussion of the properties of this equation. We discuss the behaviour of this equation as the mass of one of the two particles approaches the limitM ₂→∞ in the ladder approximation of single photon exchange. We show that up to orderO(α⁴) this equation is consistent with the Dirac equation. If the crossed two photon exchange diagrams are taken into account the difference between them is of orderO(α⁶).     

Theory of light scattering from a system of interacting Brownian particles

Starting from a N-particle diffusion equation for a system of N interacting spherical Brownian particles, a non-linear transport equation for concentration fluctuations δc(r, t) of the particles is derived. This dynamic equation is transformed into a hierarchy of equations for retarded propagators of increasing numbers of concentration fluctuations. A cluster expansion to lowest order in the average concentration results in a set of two coupled equations. The spectrum of light scattered by the interacting particles is in general not a Lorentzian, due to the non-linear term in the transport equation. For small scattering wave vectors k the width is D(ω)k², where ω is the transferred frequency. It is shown that D(0) = D_e, the effective diffusion coefficient. For a hardcore interaction potential the spectrum is Lorentzian and it is found that D_e = D₀(1 + φ), where D₀ is the diffusion constant for independent particles and φ the volume concentration of Brownian particles.     

On the connection between the macroscopical and microscopical evolution in an exactly soluble hopping model: I. Neutral particles

The hopping rate equation for neutral particles, on an arbitrary periodical lattice, can be solved exactly. It is shown that if one scales the time t and the distances x(t→λ²t, x→λx) then, in the λ→∞ limit, the particle density tends to the solution of the diffusion equation faster than λ^?3. The diffusion coefficient is the same as obtained from both Kubo and Miller-Abrahams theory via the Einstein relation.     

Relativistic remnants in the reduction of the Bethe-Salpeter equation to the Schrödinger equation

Following Salpeter, the Bethe-Salpeter equation for the bound system of two oppositely charged particles is reduced to a Schrödinger equation for each of the following cases: (a) both particles are spin 1/2 particles, (b) one particle is a spinor while the other is spinless, and (c) both particles are spinless. It is shown that ife is the magnitude of charge carried by each of the particles whose masses are set equal to the electron and proton masses then, strictly speaking, only in case (a) do we obtain the familiar Schrödinger equation for the hydrogen atom. The latter equation is recovered in the other two cases only if relativistic remnants—terms of the order of 10^?5 and smaller—are neglected in comparison with unity. Attention is drawn to a situation where such remnants may not be negligibly small, viz. the problem of confinement of quarks.     

Alpha-Particles within Nuclei

We present a formalism describing the bound state of a large number of bosons and apply it to study nuclei consisting of A α particles. The method has its roots in a few-body approach and is based on the expansion of the many-body Faddeev components in Potential Harmonics, and the subsequent reduction of the Faddeev equation into a two-variable, integro-differential equation. For A → ∞ this equation is transformed into a new simpler integro-differential equation, which is easy to use in calculations for A up to as large as 1000. We use both integro-differential equations to investigate the behavior of nuclei subject to the assumption that they are composed of α particles. Various α α forces were employed. For the Ali-Bodmer potential we found that the A = 5 system (i.e. ²⁰Ne) is the most stable, while for the A = 10 system (i.e. ⁴⁰Ca) the binding energy has a maximum. The formalism predicts α-decay for larger nuclei, but the value of A where this begins to happen is strongly dependent on the α α potential.     

A Particle System with Explosions: Law of Large Numbers for the Density of Particles and the Blow-Up Time

Consider a system of independent random walks in the discrete torus with creation-annihilation of particles and possible explosion of the total number of particles in finite time. Rescaling space and rates for diffusion/creation/annihilation of particles, we obtain a strong law of large numbers for the density of particles in the supremum norm. The limiting object is a classical solution to the semilinear heat equation ? _t u=? _xx u+f(u). If f(u)=u ^p , 1<p??3, we also obtain a law of large numbers for the explosion time.     

Effect of anomalously slow motions of particles with spins of 1/2 on the shape of the magnetic resonance line of pairs of particles

A non-Markovian version of the Liouville stochastic equation was used to analyze spin relaxation in a pair of particles with spins 1/2 and the dipole-dipole interaction between the spins. The particles were involved in anomalously slow stochastic relative rotation described by the angular correlation function K(t) ～ 1/t ^α with α < 1. The Liouville stochastic equation could be used to describe memory effects in the kinetic dependence of rotational relaxation resulting in a very slow descent of K(t). Our analysis showed that the anomalous relative motion of radicals manifested itself in special features of the shape of the magnetic resonance line of the particle pair.     

Light charged particles associated with subthreshold neutral pion emission in the16O+27Al reaction

The production ofZ=1 andZ=2 particles associated with neutral pion emission in the¹⁶O+²⁷Al reaction at 94 MeV/nucleon has been studied. Results are compared with previous findings obtained by charged pions in the same collision at the same bombarding energy and with the prediction of a dynamical model based on a numerical solution of the Boltzmann-Nordheim-Vlasov equation.     

Rotational diffusion of a tracer colloid particle: I. Short time orientational correlations

We extend the generalized Smoluchowski equation to descrbe the diffusional relaxation of position and orientation in a suspension of interacting spherical colloid particles. Considering a tracer particle which interacts with other particles through spherically symmetric pair potentials and with an external field we obtain a cluster expansion representation of the orientational time correlation functions for the tracer. The one and two body cluster contributions are studied explicitly at short times. Working to first order in volume fraction φ we show that the initial slope of the time correlation functions is described by a modified diffusion coefficient D^r = D^r₀(1 −C^rφ) where C^r is a number determined by hydrodynamic and potential interactions. We evaluate C^r numerically for spheres with slip-stick hydrodynamic boundary conditions and also for permeable spheres.     

Fractional Kinetics in Kac–Zwanzig Heat Bath Models

We study a variant of the Kac–Zwanzig model of a particle in a heat bath. The heat bath consists of n particles which interact with a distinguished particle via springs and have random initial data. As n → ∞ the trajectories of the distinguished particle weakly converge to the solution of a stochastic integro-differential equation—a generalized Langevin equation (GLE) with power-law memory kernel and driven by 1/f ^α -noise. The limiting process exhibits fractional sub-diffusive behaviour. We further consider the approximation of non-Markovian processes by higher-dimensional Markovian processes via the introduction of auxiliary variables and use this method to approximate the limiting GLE. In contrast, we show the inadequacy of a so-called fractional Fokker–Planck equation in the present context. All results are supported by direct numerical experiments.     

On the spectrum of particles created in a Robertson-Walker universe

Created particle spectra are calculated in Robertson-Walker universes with scale factora∝t ^α(0 <α≦ 1) anda∝e ^t, and discussed with a special emphasis on their dependence upon the initial and final times at which a WKB-like positive frequency condition should be imposed. It is shown that the obtained spectra are very sensitive to these times if the WKB approximation for the field equation is not valid in their neighborhood. It is also shown that the total number density of created particles remains finite if the final time is set to be finite.     

On the vacuum rearrangement in massless chromodynamics

It is shown that in the weak coupling limit the Bethe-Salpeter equation of massless chromodynamics admits a colourless tachyon solution when the number of quark multiplets n ? n_c, where for the colour group SU(N) the critical value n_c ≈ 0.14 N³/(N² ? 1). When n ? n_c, the vacuum rearrangement results in the gluons acquiring a mass; n > n_c all particles remain massless.     

Approximate Treatment of 3-Body Coulomb Systems: Discrete Spectrum

We present a method for treatment of three charged particles. The proposed method has universal character and is applicable both for bound and continuum states. A finite-rank approximation is used for Coulomb potential in the Lippman?CSchwinger equation, that results in a system of one-dimensional coupled integral equations. Preliminary numerical results for three-body atomic and molecular systems like H ^?, He, pp?? and other are presented.     

Field theory,curdling, limit cycles,and cellular automata

It is suggested that the process of curdling is an important question for the science of fractals. A field equation which displays nucleation (curdling) of particles out of a pure radiation field is discussed. The particle formation arises naturally from the nonlinear character of the equation rather than from imposed quantization conditions. The relativistically invariant equation is $$div(\rho ^\mu (r,t,\Omega _1 )) = \int {[\rho _\mu (r,t,\Omega ),\rho ^\mu (r,t,\Omega _2 )]d} \Omega _2 $$ where ¦, ¦ denotes commutator.ρ ^μ (r,t,Ω) is both a 4-vector and a 2×2 matrix. It represents substance atr, t traveling with the velocity of light in direction Ω. A unique feature is that the scattering ofρ(Ω ₁) byρ(Ω ₂) as determined by the right-hand side of the above equation results in fields that persist at a given place even thoughρ itself represents substance traveling always at the speed of light. Explicit solutions are given for the case of one dimension. Fields representing particles are obtained and shown to have specially oscillatory structure with incipient fractal character.     

One-dimensional hard rod caricature of hydrodynamics

We give here a rigorous deduction of the “hydrodynamic” equation which holds in the hydrodynamic limit, for a model system of one-dimensional identical hard rods interacting through elastic collisions. The equation should be considered as the analog of the Euler equation of real hydrodynamics. Owing to the degeneracy of the model, it is written in terms of a functiong(q, v, t) expressing the density of particles with velocityv at the pointq at timet. For this equation we prove an existence and uniqueness theorem in some natural class of functions. Our main result is the proof that if {^∈, ∈ >0} is a class of initial states which are homogeneous on a scale much less than ε^?1, and if the corresponding particle densities tend, asε→0, in the proper scale, to the initial hydrodynamic densityg _o (q,v), then, under some general assumptions on the states ∈^? and ong ₀, the particle densities of the evolved states at timeε ^?1 t, tend asε→0 to the unique solution of the hydrodynamic equation with initial conditiong ₀. The proof is completed by exhibiting a large class of initial families {^∈, ∈ >0} which possess the required properties.     

Probability and complex quantum trajectories

It is shown that in the complex trajectory representation of quantum mechanics, the Born’s ΨΨ probability density can be obtained from the imaginary part of the velocity field of particles on the real axis. Extending this probability axiom to the complex plane, we first attempt to find a probability density by solving an appropriate conservation equation. The characteristic curves of this conservation equation are found to be the same as the complex paths of particles in the new representation. The boundary condition in this case is that the extended probability density should agree with the quantum probability rule along the real line. For the simple, time-independent, one-dimensional problems worked out here, we find that a conserved probability density can be derived from the velocity field of particles, except in regions where the trajectories were previously suspected to be nonviable. An alternative method to find this probability density in terms of a trajectory integral, which is easier to implement on a computer and useful for single particle solutions, is also presented. Most importantly, we show, by using the complex extension of Schrodinger equation, that the desired conservation equation can be derived from this definition of probability density.     

Newtonian versus Langevin dynamics: Example of ϕ4-model with infinite range interactions

The functional integral representation for the generating functional (GF) of the canonically averaged ensemble with an underlying Newtonian dynamics is obtained. It is shown that for this representation the non-linear fluctuation-dissipation theorem (NFDT) has the same form as for the Langevin dynamics case. This GF-representation is used for the investigation of the dynamics of the ϕ⁴-model with infinite range interactions at T > T_c. It is shown that the kinetic equation for the complete correlation function has the same form as for the Langevin dynamics case, which was considered before. All peculiarities of Newtonian dynamics are absorbed by one-particle (2-point and 4-point) correlator and response functions. The analysis of this equation shows that the 1/N-fluctuations (where N is the number of particles) restore the ergodicity of the system with the characteristicsrate τ⁻¹ ∼ μ²/N, where μ is a coupling constant.